Method and apparatus for fabricating a light management substrates

ABSTRACT

A method of machining a surface of a workpiece is accomplished by bringing a cutting tool into contact with the surface of the workpiece and for at least one cutting pass, i, causing relative movement between the cutting tool and the surface of the workpiece along a path in the surface of the workpiece. The path is in the nature of a mathematical function defined over a segment, C, of a coordinate system and characterized by a set of nonrandom, random or pseudorandom parameters selected from the group consisting of amplitude, phase and period or frequency. Relative movement between the cutting tool and the surface of the workpiece may be accomplished by bandpass filtering a noise signal; providing the bandpass filtered signal to a function generator; generating a randomly modulated mathematical function from the function generator; and in response to the randomly modulated function, directing the relative movement between the cutting tool and the surface of the workpiece along the path in the surface of the workpiece.

BACKGROUND OF INVENTION

[0001] This invention relates to a method and apparatus for fabricatinglight management films and in particular to such films fabricated fromrandomly or pseudo randomly mastered surfaces.

[0002] In backlight computer displays or other systems, films arecommonly used to direct or diffuse light. For example, in backlightdisplays, brightness enhancement films use prismatic or texturedstructures to direct light along the viewing axis (i.e., normal to thedisplay), which enhances the brightness of the light viewed by the userof the display and which allows the system to use less power to create adesired level of on-axis illumination. Films for turning light can alsobe used in a wide range of other optical designs, such as for projectiondisplays, traffic signals, and illuminated signs. Backlight displays andother systems use layers of films stacked and arranged so that theprismatic or textured surfaces thereof are perpendicular to one anotherand are sandwiched between other optical films known as diffusers.Diffusers have highly irregular or randomized surfaces.

[0003] Textured surfaces have been widely used in optical applicationssuch as backlight display films, diffusers, and rear reflectors. In awide range of optical designs it is necessary to use microstructures toredirect and redistribute light (or diffuse light) to enhancebrightness, diffusion, or absorption. For example, in a backlightdisplay system it is often required to both direct the illuminationincident on a screen toward a direction normal to the screen and tospread the illumination over the viewer space. Performance of thin-filmsolar cells can be markedly improved by light trapping based on texturedTCO/glass/metal substrates, and angle selective specular reflectors.Microstructures are sometimes randomized for reducing manufacturingdefects such as pits and defects from optical interference between twocomponents such as moir épattern, speckle and Newton's ring. Ideally, anoptical film, instead of two or more films together, should have boththe performance of brightness enhancement and least defects.

[0004] In backlight applications brightness enhancement films anddiffuser films are commonly combined as part of a display screen toredirect and redistribute light. In the prior art a typical solution forenhancing brightness is to use an optical film having a surfacestructured with linear prisms. For example, the prior art describesusing a prismatic film to enhance the on-axis brightness of a backlightliquid crystal display. To hide manufacturing defects and decrease theoptical coupling, an optical film with structures randomly varying inheight along its length direction has been crafted to achieve brightnessenhancement while hiding manufacturing defects and reducing opticalcoupling between two sheets of such film.

SUMMARY OF INVENTION

[0005] A method of machining a surface of a workpiece is accomplished bybringing a cutting tool into contact with the surface of the workpieceand for at least one cutting pass, i, causing relative movement betweenthe cutting tool and the surface of the workpiece along a path in thesurface of the workpiece. The path is in the nature of a mathematicalfunction defined over a segment, C, of a coordinate system andcharacterized by a set of nonrandom, random or pseudorandom parametersselected from the group consisting of amplitude, phase and period orfrequency.

[0006] Relative movement between the cutting tool and the surface of theworkpiece may be accomplished by bandpass filtering a noise signal;providing the bandpass filtered signal to a function generator;generating a randomly modulated mathematical function from the functiongenerator; and in response to the randomly modulated function, directingthe relative movement between the cutting tool and the surface of theworkpiece along the path in the surface of the workpiece.

[0007] The invention works by modulating the prism structures of anoptical film from the nominal linear path in a lateral direction(direction perpendicular to the height) by using a nonrandom, random (orpseudo random) amplitude and period. Masters for the tools tomanufacture films having such microstructures may be made by diamondturning on a cylindrical drum or flat plate. The drum is typicallycoated with hard copper or Nickel upon which the grooves are eitherthread or annular cut. The drum is turning while the diamond cuttingtool is moving transverse to the turning direction for a thread cut oran annular cut to produce the desired pitch. In order to produce themodulation, a fast tool servo (FTS) system is used to drive the toollaterally. A piezoelectric transducer is used to move the diamond toolto a desired displacement by varying the voltage supplied to thetransducer at a random or pseudo random frequency. Both the displacement(amplitude) and the frequency at any instant can be randomly generatedin a personal computer and then sent to the amplifier to produce thedesired voltage. Because of temperature and hysteresis effects of thepiezoelectric materials, a feed back control with a distance probe maybe required to ensure the correct tool movement. For modulating the cutin both lateral direction and height, a FTS with two transducers withindependent controllers and probes may be used.

[0008] The invention reduces the number of components in an opticalsystem and thus reduces cost and weight. Generally it improves opticalperformance by minimizing many conceivable optical interferences andcouplings. The manufacturing methods provide microstructures with morecontrol over the light direction.

[0009] The invention provides light enhancement and diffusion withoutoptical artifacts by randomly varying the prism structures in a lateraldirection and height. Because of the random component in the lateraldirection, the optical defects resulting from interference between twosheets of optical films such as Moir é patterns, speckles and Newton'srings are almost absent. The lateral variation is more effective thanthe height variation in producing diffusion and reducing optical defectsespecially the Moire effect. The randomness of the prism patterns allowsblending the joints of machined patches without visible seams. Thelength of the drum is thus not limited by the cutting tool travel.Lateral motion of the cutting tool has feedback control to ensureprecise positioning to overcome hysteresis, creep and temperature topiezoelectric stack. The combination of lateral and height variationsprovides greater freedom in machining surface microstructures for manyapplications such as diffusers, solar cell panels, reflectors.

BRIEF DESCRIPTION OF DRAWINGS

[0010]FIG. 1 is a flow chart showing a method of machining a surface ofa workpiece wherein the workpiece is a master drum;

[0011]FIG. 2 is a flow chart showing a method of machining a surface ofa workpiece wherein the workpiece is on a master plate;

[0012]FIG. 3 is a diagram of the master drum of FIG. 1 having a randomor pseudo random pattern therein following a generally spiral-like orthreaded path;

[0013]FIG. 4 is a diagram of the master drum of FIG. 1 having a randomor pseudo random pattern therein over generally concentric rings;

[0014]FIG. 5 is a diagram of the master plate of FIG. 2 having a randomor pseudo random pattern therein following a generally sawtooth ortriangular path;

[0015]FIG. 6 is a diagram of the master plate of FIG. 2 having a randomor pseudo random pattern therein along a series of generally concentricrings;

[0016]FIG. 7 is a diagram of a cross section of a cutting tool in thenature of a prismatic structure;

[0017]FIG. 8 is a diagram of the prismatic cutting tool of FIG. 5 havingcompound angled facets;

[0018]FIG. 9 is a graphical representation of the magnitude of the powerspectral density of the randomized surface of the workpiece as afunction of spatial frequency;

[0019]FIG. 10 is a top view of the randomized surface of the workpiecegenerated by the method of the invention;

[0020]FIG. 11 is a graphical representation of a plurality of paths dueto a plurality of cutting passes over the surface of the workpiece;

[0021]FIG. 12 is a schematic representation of a system and apparatusfor machining the surface of a workpiece in communication over acommunications or data network with remote locations;

[0022]FIG. 13 is a graphical representation of mathematical functions;

[0023]FIG. 14 is a three dimensional view of a backlight display device;

[0024]FIG. 15 is a schematic diagram of a master machining system with afast tool servo for cutting grooves having lateral variations in thesurface of a workpiece; and

[0025]FIG. 16 is a depiction of a cutting gradient introduced into thesurface of the machined surface of the workpiece.

DETAILED DESCRIPTION

[0026] Referring to FIG. 1, a method of machining a surface of a workpiece is shown generally at 100. A noise signal 102 is band passfiltered 104 and provided as input to a function generator 106. Amodulated mathematical function, such as a sinusoidal wave form isprovided by the function generator 106 as input to a servo mechanism108. The noise signal 102, the bandpass filter 104 and the functiongenerator 106 can be replaced by a computer system equipped with theappropriate signal processing software and digital-to-analog conversionboard so as to generate the input signal to the servo mechanism 108. Theservo mechanism 108 directs relative movement between a cutting tool 110and the surface of a drum 112 rotating at an angular velocity of ω in acylindrical coordinate system (r, θ,z). As the drum 112 rotates atangular velocity ω, the cutting tool 110 moves relative to the drum 112along the drum axis, z, and randomly moves back and forth with afrequency of up to about 2,000 Hz parallel to the axis of the drum 112The cutting tool 110, being in continuous contact with the surface ofthe rotating drum 110, thus cuts or machines a randomized spiral-like orthreaded pattern 116 (FIG. 3) having a pitch, P, into the surface of thedrum 112. For a two axis cutting tool 110, the cutting tool moves notonly back and forth parallel to the drum axis 112, but alsoperpendicular to the drum surface to cut different depths in the surfaceof the drum 112.

[0027] Alternatively, as seen in FIG. 2, the cutting tool 110 may be incontact with the surface of a flat plate 114 moving at a velocity of vin a rectilinear coordinate system (x,y,z). Similarly, as the plate 114moves at velocity v, and the cutting tool 110, randomly moves back andforth across the plate, the cutting tool 110, being in continuouscontact with the surface of the plate 114, thus or machines a randomizedtriangular pattern 122 (FIG. 5) into the surface of the plate 114.

[0028] In an alternative embodiment of the invention, as seen in FIG. 4,the drum 112 need not move along the z axis as the drum 112 rotates. Assuch, the cutting tool machines a randomized or pseudo randomizedpattern along a series of i concentric rings 118 in the surface of thedrum 112 whereby the cutting tool returns to a starting point 122 foreach cutting pass. To achieve good cutting quality, a control system canallow the cutting tool 110 to repeat the pattern of any i^(th) cuttingpass for the number of revolutions depending upon the desired final cutdepth and in-feed rate. When the cutting tool 110 finishes the number ofrevolutions and returns to the starting point 122 of the i^(th) cuttingpass, the cutting tool 110 is shifted or stepped a distance S_(i) to thenext, or k^(th), cutting pass.

[0029] It will be understood that the cutting tool 110 may have morethan one axis of travel. For example it can have three axes of travel r,θ, z in cylindrical coordinates and x, y, z in rectilinear coordinates.Such additional axes will allow for the cutting of toroidal lens typestructures when using a radiused cutting tool 110 or allow for agradient in the cut along the cut length, for example. Translationalaxes r, θ, z and x, y, z will also allow for introducing a cuttinggradient into the pattern machined into the surface of the workpiece112, 114 for subsequent cutting passes. Such a cutting gradient is bestseen with reference to FIG. 16. In FIG. 16, the i^(th) cutting pass hasa thickness or width of w_(i) and the k^(th) cutting pass has athickness of w_(k) where w_(i) is greater or less than w_(k).Furthermore, the n^(th) cutting pass has a width of w_(n) where w_(n) isgreater or less than w_(k). It will be understood that the change in thethickness in the cutting pattern in subsequent cutting passes may benonrandom, random or pseudo random. Additional rotational degrees offreedom (e.g., pitch 152, yaw 150 and roll 154, FIGS. 1, 2, 5 and 6) maybe used to change the angular orientation of the cutting tool 110 withrespect to the surface of the workpiece 112, 114, thus changing thegeometry of the facets machined into the master surface.

[0030] The randomized or pseudo randomized pattern machined into thesurface of the work piece 112, 114 is in the nature of a mathematicalfunction defined over a segment, C, of a coordinate system andcharacterized by a set of random or pseudorandom parameters selectedfrom the group consisting of amplitude, phase and frequency. For arotating drum 112 the segment, C, which the mathematical function isdefined is the circumference of the drum 112. For a moving plate 114,the segment, C, over which the mathematical function is defined is awidth or length of the plate 114. An exemplary mathematical function isthat of the sine wave of Equation 1:

y _(i) =A _(i) sin{Ψ_(i) }+S _(i)  (1)

[0031] wherein y_(i) is the instantaneous displacement of the cuttingtool relative to C on the i^(th) cutting pass, A_(i) is the displacementof the cutting tool relative to C,

Ψ_(i)|

[0032] is the phase of y_(i) and S_(i) is a shift in the startingposition of y_(i).

[0033] In Eq. (1), the phase,

Ψ_(i)|

[0034] , is $\begin{matrix}{\psi_{i} = {{\varphi ( {\lambda_{i} - \frac{q_{i + k}}{2}} )} - \Phi_{i}}} & (2)\end{matrix}$

[0035] where Ø is a number between zero and 2 π radians inclusive. Inorder for the cutting tool 110 to return to the starting position 122 atthe end of the i^(th) cutting pass, the segment, C, over which themathematical function is defined is equal to an integer number of halfwavelengths, λ_(i). Thus, for the i^(th) cutting pass: $\begin{matrix}{{\frac{N}{2}\lambda_{i}} = {C\quad \text{or}}} & (3)\end{matrix}(4)$ $\lambda_{i} = \frac{2 \times C}{N}$

[0036] where N is a randomly or pseudo randomly chosen positive ornegative integer. In Eq. (2), $\frac{q_{i + k}}{2}$

[0037] is an additive factor modifying λ_(i) on a k^(th) subsequentcutting pass and

Φ_(i)

[0038] is a randomly or pseudo randomly chosen number between zero and 2π radians inclusive.

[0039] Yet further in Eq. (1), the phase,

Ψ_(i)|

[0040] may be: $\begin{matrix}{\psi_{i} = {{\varphi ( {\lambda_{i} - \frac{q_{i + k}}{2}} )} - ~\Phi_{i} - {a_{i}\sin \{ {\varphi ( {\frac{\lambda_{i}}{b_{i}} - \Omega_{i}} )} \}}}} & (5)\end{matrix}$

[0041] where a_(i) and b_(i) are scalar quantities and Ω_(i) is anonrandomly, randomly or pseudo randomly chosen number between zero and2 π radians inclusive.

[0042] It will be understood that the mathematical function referred toabove may be any mathematical function that can be programmed into acomputer numerically controlled (CNC) machine. Such functions includefor example the well known triangular function, sawtooth function andsquare wave function (FIG. 13) each of which may be randomly modulatedin amplitude, phase and frequency.

[0043] Referring to FIGS. 7 and 8, the cutting tool 110 comprises aprismatic structure having a cross section which may include straightfacets 130, 132 intersecting at a tip 134 at a peak angle of 2 θ. Theprismatic shaped cutting tool 110 may also comprise linear segments 138,140 of the facets 132, 134 resulting in a compound angled prism. Thecompound angle prism has a first facet 138 at an angle of α and a secondfacet 140 at an angle of β with respect to a base 142 of the prism 110.As best understood from FIGS. 7 and 8, the cutting tool 110 may have across section with a rounded peak 134 or radius “r.” In general thecutting tool can have a cross section of any manufacturable shape.

[0044] The equipment needed to machine the surface of the workpiece112,114 in the invention is shown in FIG. 12. Machining the surface ofthe workpiece 112, 114 is accomplished by following a method thatutilizes a computer numerically controlled (CNC) milling or cuttingmachine 202, having a cutting tool 110, which is controlled by asoftware program 208 installed in a computer 204. The software program208 is written to control the movement of the cutting tool 110. Thecomputer 204 is interconnected to the CNC milling machine 202 with anappropriate cabling system 206. The computer 204 includes a storagemedium 212 for storing the software program 208, a processor forexecuting the program 208, a keyboard 210 for providing manual input tothe processor, and a modem or network card for communicating with aremote computer 216 via the Internet 214 or a local network.

[0045] In FIG. 15, a master machining system with a fast tool servo forcutting grooves having lateral variations in the surface of a workpieceis shown generally at 400. An input/output data processor 402 providescutting commands to a digital signal processing (DSP) unit 404 whichsupplies a signal to a digital-to-analog (DA) conversion device 406. Avoltage amplifier 408 receptive of a signal from the DA converter 406drives a fast tool servo mechanism 410 to direct the motion of thecutting tool 110. A cutting tool position probe 412 senses the positionof the cutting tool 110 and provides a signal indicative of cutting toolposition to a sensor amplifier 418 which amplifies the signal. Theamplified signal is directed to an analog-to-digital (A/D) converter420. A lathe encoder 414 determines the position of the workpiece (e.g.,drum 112) and provides a feedback signal to the A/D converter 420. TheA/D converter thus provides as output to the digital signal processingunit 404, a feedback signal indicative of the position of the cuttingtool 110 and the position of the workpiece 112,114 and the DSP unit 404provides a processed signal to the input/output processor 402.

[0046] Following the method results in a randomly or pseudo randomlymachined surface of the workpiece 112, 114. When the computer 204,having the software program 208 installed, is in communication with theCNC milling machine 202 an operator is ready to begin the method thatwill randomly or pseudo randomly machine the surface of the workpiece112, 114. Following the method, the operator begins by providing asinput the values of A_(i) and

Ψ_(i)|

[0047] into the personal computer 204. The operator input can beprovided manually by typing the values for A_(i) and

Ψ_(i)|

[0048] using the keyboard 210. The mathematical function or functions(FIG. 13) may be stored within the computer's memory or may be stored ona remote computer 216 and accessed via the Internet 214 or via a localnetwork.

[0049] The operator is prompted to provide the values of A and into theCNC machine 202. Once these values are provided, the cutting element 110of the CNC machine 202 begins to mill the workpiece 112, 114. Thecommands provided by the software program 208 will precisely direct thecutting tool 110 to mill the workpiece 112, 114 accordingly. This isachieved by the monitoring of movement of the cutting tool 110 in anappropriate coordinate system. Additionally, the program 208 preciselycontrols the depth to which the milling process occurs. This is alsoachieved by the monitoring of the movement of the cutting tool 110 inthe coordinate system. As best understood, the nonrandomized, randomizedor pseudo randomized surface of the workpiece 112, 114 resulting fromthe cutting process may be in the form of a“positive” or a“negative”master.

[0050] From the master, an optical substrate 142 (FIG. 10) may begenerated by forming a negative or positive electroform over the surfaceof the workpiece 12, 114. Alternatively, a molding material can be usedto form a replica of the original positive or negative master—forexample, a ultraviolet (UV) or thermal curing epoxy material or siliconmaterial. Any of these replicas may be used as a mold for a plasticpart. Embossing, injection molding, or other methods may be used to formthe parts.

[0051] In surface metrology the autocorrelation function, R(x,y), is ameasure of the randomness of a surface. Over a certain correlationlength, Ic, however, the value of an autocorrelation function, R (x,y),drops to a fraction of its initial value. An autocorrelation value of1.0, for instance, would be considered a highly or perfectly correlatedsurface. The correlation length, Ic, is the length at which the value ofthe autocorrelation function is a certain fraction of its initial value.Typically, the correlation length is based upon a value of 1/e, or about37 percent of the initial value of the autocorrelation function. Alarger correlation length means that the surface is less random than asurface with a smaller correlation length.

[0052] In some embodiments of the invention, the value of theautocorrelation function for the three-dimensional dimensional surfaceof the optical substrate 142 drops to less than or equal to 1/e of itsinitial value in a correlation length of about 1 cm or less. In stillother embodiments, the value of the autocorrelation function drops to1/e of its initial value in about 0.5 cm or less. For other embodimentsof the substrate the value of the autocorrelation function along thelength I drops to less than or equal to 1/e of its initial value inabout 200 microns or less. For still other embodiments, the value of theautocorrelation function along the width w drops to less than or equalto 1/e of its initial value in about 11 microns or less.

[0053] In FIG. 14 a perspective view of a backlight display 300 deviceis shown. The backlight display device 300 comprises an optical source302 for generating light 306. A light guide 304 guides the light 306therealong by total internal reflection (TIR). The light guide 304contains disruptive features that cause the light 306 to escape thelight guide 304. Such disruptive features may include, for example, asurface manufactured from a master having a cutting gradient machinedtherein as explained with regard to FIG. 16. A reflective substrate 308positioned along the lower surface of the light guide 304 reflects anylight 306 escaping from the lower surface of the light guide 304 backthrough the light guide 304 and toward an optical substrate 314fabricated from a positive or negative master having a nonrandomized;randomized or pseudo randomized surface. At least one optical substrate314 is receptive of the light 306 from the light guide 304. The opticalsubstrate 314 comprises on one side thereof a planar surface 310 and ona second opposing side thereof a randomized surface 312 generated fromthe randomized surface of the workpiece 112, 114 (e.g., the master drumor master plate). The optical substrate 314 may also comprise arandomized surface of both sides thereof. The optical substrate 314 isreceptive of the light 306 and acts to turn and diffuse the light 306 ina direction that is substantially normal to the optical substrate 314along a direction z as shown. The light 306 is then directed to an LCDfor display. A diffuser 316 may be located above the optical substrate314 to provide diffusion of the light 306. This substrate 314 may be aretarder film that is used to rotate the plane of polarization of thelight exiting the optical substrate 314 such that the light is bettermatched to the input polarization axis of an LCD. A half wave retarder,for example, may be used to rotate the substantially linearly polarizedlight exiting the optical substrate 314. The retarder may be formed bystretching a textured or untextured polymer substrate along one axisthereof in the plane of the substrate. Alternatively, a liquid or solidcrystal device may be used. Alternatively, for this purpose the retarderfilm 316 could be built into the lower LCD substrate.

[0054] Any references to first, second, etc. or front and back, rightand left, top and bottom, upper and lower, and horizontal and vertical,or any other phrase that relates one variable or quantity with respectto another are, unless noted otherwise, intended for convenience ofdescription, not to limit the present invention or its components to anyone positional or spatial orientation. All dimensions of the componentsin the attached Figures can vary with a potential design and theintended use of an embodiment without departing from the scope of theinvention.

[0055] While the invention has been described with reference to severalembodiments thereof, it will be understood by those skilled in the artthat various changes may be made and equivalents may be substituted forelements thereof without departing from the scope of the invention. Inaddition, many modifications may be made to adapt a particular situationor material to the teachings of the invention without departing from theessential scope thereof. Therefore, it is intended that the inventionnot be limited to the particular embodiments disclosed as the best modecontemplated for carrying out this invention, but that the inventionwill include all embodiments falling within the scope of the appendedclaims.

1. A method of machining a surface of a workpiece, the methodcomprising: bringing a cutting tool into contact with the surface of theworkpiece; and for at least one cutting pass, i, causing relativemovement between the cutting tool and the surface of the workpiece alonga path in the surface of the workpiece; wherein the path is in thenature of a mathematical function defined over a segment, C, of acoordinate system and characterized by a set of nonrandom, random orpseudorandom parameters selected from the group consisting of amplitude,phase and frequency.
 2. The method as set forth in claim 1 wherein themathematical function is defined by the equation y _(i) =A _(i)sin{Ψ_(i)}=S _(i) wherein i is an integer indicative of the number of the cuttingpath, yi is the instantaneous displacement of the cutting tool relativeto C on the ith cutting pass, Ai is the maximum displacement of thecutting tool relative to C, Ψ_(i)| is the phase of yi and Si is a shiftin the starting position of yi.
 3. The method as set forth in claim 2wherein$\psi_{i} = {{\varphi ( {\lambda_{i} - \frac{q_{i + k}}{2}} )} - \Phi_{i}}$

where φ is a number between zero and 2π inclusive,$\lambda_{i} = \frac{2 \times C}{N}$

where N is a nonrandomly, randomly or pseudo randomly chosen positive ornegative integer, $\frac{q_{i + k}}{2}$

is an additive factor modifying λ_(i) on a k^(th) subsequent cuttingpass and Φ_(i) is a nonrandomly, randomly or pseudo randomly chosennumber between zero and 2 π inclusive.
 4. The method as set forth inclaim 2 further comprising, for a kth cutting pass, subsequent to theith cutting pass, shifting the starting point of the mathematicalfunction a distance Si from the starting point of the mathematicalfunction on the ith cutting pass.
 5. The method as set forth in claim 2further comprising nonrandomly, randomly or pseudo randomly assigning avalue to Ai.
 6. The method as set forth in claim 1 wherein themathematical function is selected from the group of mathematicalfunctions consisting of triangular function, sawtooth function andsquare wave function.
 7. The method as set forth in claim 2 wherein$~{\psi_{i} = {{\varphi ( {\lambda_{i} - \frac{q_{i + k}}{2}} )} - \Phi_{i} - {\alpha_{i}\quad \sin \{ {\varphi ( {\frac{\lambda_{i}}{b_{i}} - \Omega_{i}} )} \}}}}$

where φ is a number between zero and 2π inclusive,$\lambda_{i} = \frac{2 \times C}{N}$

where N is a nonrandomly, randomly or pseudo randomly chosen positive ornegative integer, $\frac{q_{i + k}}{2}$

is an additive factor modifying Ψ_(i)| on a kth subsequent cutting passand Φ_(i) is a randomly or pseudo randomly chosen number between zeroand 2 π inclusive, ai and b are scalar quantities and Ω i is anonrandomly, randomly or pseudo randomly chosen number between zero and2 π inclusive.
 8. The method as set forth in claim 7 further comprising,for a kth cutting pass, subsequent to the ith cutting pass, shifting thestarting point of the mathematical function a distance Si from thestarting point of the mathematical function on the ith cutting pass. 9.The method as set forth in claim 6 further comprising nonrandomly,randomly or pseudo randomly assigning a value to Ai.
 10. The method asset forth in claim 1 wherein causing relative movement between thecutting tool and the surface of the workpiece includes: bandpassfiltering a noise signal; providing the bandpass filtered noise signalto a function generator; generating a randomly modulated mathematicalfunction from the function generator; in response to the randomlymodulated function, directing the relative movement between the cuttingtool and the surface of the workpiece along the path in the surface ofthe workpiece.
 11. A method of fabricating an optical substrate from amachined surface of a workpiece, the method comprising: forming apositive or negative electroform over the surface of the surface of theworkpiece; forming a replica of the electroform; and transferring thereplica of the electroform to an optical substrate.
 12. An opticalsubstrate comprising: a surface formed from a master surface machined bybringing a cutting tool into contact with a surface of a workpiece; forat least one cutting pass, i, causing relative movement between thecutting tool and the surface of the workpiece along a path in thesurface of the workpiece; wherein the path is in the nature of amathematical function defined over a segment, C, of a coordinate systemand characterized by a set of nonrandom, random or pseudorandomparameters selected from the group consisting of amplitude, phase andfrequency; forming a positive or negative electroform over the surfaceof the surface of the workpiece; forming a replica of the electroform;and transferring the replica of the electroform to an optical substrate.13. A backlight display device comprising: an optical source forgenerating light; a light guide for guiding the light therealong; areflective device positioned along the light guide for reflecting thelight out of the light guide; an optical substrate receptive of thelight from the light guide, the optical substrate comprising: a surfaceformed from a master surface machined by bringing a cutting tool intocontact with a surface of a workpiece; for at least one cutting pass, i,causing relative movement between the cutting tool and the surface ofthe workpiece along a path in the surface of the workpiece; wherein thepath is in the nature of a mathematical function defined over a segment,C, of a coordinate system and characterized by a set of nonrandom,random or pseudorandom parameters selected from the group consisting ofamplitude, phase and frequency; forming a positive or negativeelectroform over the surface of the surface of the workpiece; forming areplica of the electroform; and transferring the replica of theelectroform to an optical substrate.
 14. A master having a surfacemachined by the method as set forth in claim
 1. 15. The method as setforth in claim 4 wherein for a kth cutting pass, subsequent to the ithcutting pass, a width wk of the kth cutting pass is different than forthe ith cutting pass.
 16. The method as set forth in claim 15 whereinfor a kth cutting pass, subsequent to the ith cutting pass, thedifference in the width wk of the kth cutting pass with respect to theith cutting pass is a gradient over all cutting passes.
 17. The methodas set forth in claim 15 wherein for a kth cutting pass, subsequent tothe ith cutting pass, the difference in the width wk of the kth cuttingpass with respect to the ith cutting pass is random or pseudo random.18. A backlight display device comprising: an optical source forgenerating light; a light guide for guiding the light therealong; and areflective device positioned along the light guide for reflecting thelight out of the light guide; wherein the light guide includes a surfaceformed from a master surface machined by bringing a cutting tool intocontact with a surface of a workpiece; for at least one cutting pass, i,causing relative movement between the cutting tool and the surface ofthe workpiece along a path in the surface of the workpiece; wherein thepath is in the nature of a mathematical function defined over a segment,C, of a coordinate system and characterized by a set of nonrandom,random or pseudorandom parameters selected from the group consisting ofamplitude, phase and frequency; forming a positive or negativeelectroform over the surface of the surface of the workpiece; forming areplica of the electroform; and transferring the replica of theelectroform to an optical substrate.
 19. The display device as set forthin claim 18 wherein for a kth cutting pass, subsequent to the ithcutting pass, a width wk of the kth cutting pass is different than theith cutting pass.
 20. The display as set forth in claim 19 wherein for akth cutting pass, subsequent to the ith cutting pass, the difference inthe width wk of the kth cutting pass with respect to the ith cuttingpass is a gradient over all cutting passes.
 21. The display as set forthin claim 19 wherein for a kth cutting pass, subsequent to the ithcutting pass, the difference in the width wk of the kth cutting passwith respect to the ith cutting pass is random or pseudo random.
 22. Amethod of machining a surface of a cylinder, the method comprising:bringing a cutting tool into contact with the surface of the cylinder;and for at least one cutting pass, i, causing relative movement betweenthe cutting tool and the surface of the cylinder along a path in a planetangential to the surface of the cylinder; wherein the path is in thenature of a mathematical function defined over a segment, C, of thecylinder and characterized by a set of nonrandom, random or pseudorandomselected from the group consisting of amplitude, phase and frequency.